Abstract

The Schrage equation is commonly used in thermofluid engineering to model high-rate liquid–vapor phase change of pure fluids. Although shortcomings of this simple model were pointed out decades ago and more rigorous models have emerged from the kinetic theory community, Schrage's equation continues to be widely used. In this paper, we quantify the accuracy of the Schrage equation for evaporation and condensation of monatomic and polyatomic fluids at the low to moderately high flux operating conditions relevant to thermofluid engineering applications. As a high-accuracy reference, we numerically solve a Bhatnagar, Gross, and Krook (BGK)-like a model equation for polyatomic vapors that have previously been shown to produce accurate solutions to the Boltzmann transport equation. We observe that the Schrage equation overpredicts heat/mass fluxes by ∼15% for fluids with accommodation coefficients close to unity. For fluids with smaller accommodation coefficients, such as water, the Schrage equation yields more accurate flux estimates. We find that the Mott-Smith-like moment methods developed for liquid–vapor phase change are much more accurate than the Schrage equation, achieving heat/mass flux estimates that deviate by less than 1% (evaporation) and 4% (condensation) from the reference solution. In light of these results, we recommend using the moment method equations instead of the Schrage equation. We also provide tables with our high-accuracy numerical data for evaporation of any fluid and condensation of saturated water vapor, engineering equations fit our data, and code for moment method calculations of evaporation and condensation.

References

1.
Schrage
,
R. W.
,
1954
,
A Theoretical Study of Interphase Mass Transfer
,
Columbia Press
, New York.
2.
Hertz
,
H.
,
1882
, “
Ueber Die Verdunstung Der Flüssigkeiten, Insbesondere Des Quecksilbers, Im Luftleeren Raume
,”
Ann. Phys. Chem.
,
253
(
10
), pp.
177
200
.10.1002/andp.18822531002
3.
Knudsen
,
M.
,
1915
, “
Die Maximale Verdampfunggeschwindigkeit Des Quecksilbers
,”
Ann. Phys. Chem.
,
352
(
13
), pp.
697
708
.10.1002/andp.19153521306
4.
Kucherov
,
R. Y.
, and
Rikenglas
,
L. E.
,
1960
, “
On Hydrodynamic Boundary Conditions With Evaporation of a Solid Absorbing Radiant Energy
,”
Sov. Phys. JETP
,
37
, pp.
10
11
.
5.
Barrett
,
J.
, and
Clement
,
C.
,
1992
, “
Kinetic Evaporation and Condensation Rates and Their Coefficients
,”
J. Colloid Interface Sci.
,
150
(
2
), pp.
352
364
.10.1016/0021-9797(92)90205-Z
6.
Anisimov
,
S. I.
,
1968
, “
Metal Evaporation by Laser Radiation
,”
J. Eksperim. Teor. Fis.
,
54
, pp.
339
342
.
7.
Ytrehus
,
T.
,
1977
, “
Theory and Experiments on Gas Kinetics in Evaporation
,”
Rarefied Gas Dynamics
, American Institute of Aeronautics and Astronautics,
New York
, pp.
1197
1212
.
8.
Pao
,
Y.
,
1971
, “
Application of Kinetic Theory to the Problem of Evaporation and Condensation
,”
Phys. Fluids
,
14
(
2
), p.
306
.10.1063/1.1693429
9.
Sone
,
Y.
, and
Onishi
,
Y.
,
1978
, “
Kinetic Theory of Evaporation and Condensation –Hydrodynamic Equation and Slip Boundary Condition
,”
J. Phys. Soc. Jpn.
,
44
(
6
), pp.
1981
1994
.10.1143/JPSJ.44.1981
10.
Cercignani
,
C.
,
Fiszdon
,
W.
, and
Frezzotti
,
A.
,
1985
, “
The Paradox of the Inverted Temperature Profiles Between an Evaporating and a Condensing Surface
,”
Phys. Fluids
,
28
(
11
), p.
3237
.10.1063/1.865373
11.
Yu
,
J.
, and
Wang
,
H.
,
2012
, “
A Molecular Dynamics Investigation on Evaporation of Thin Liquid Films
,”
Int. J. Heat Mass Transfer
,
55
(
4
), pp.
1218
1225
.10.1016/j.ijheatmasstransfer.2011.09.035
12.
Liang
,
Z.
,
Biben
,
T.
, and
Keblinski
,
P.
,
2017
, “
Molecular Simulation of Steady-State Evaporation and Condensation: Validity of the Schrage Relationships
,”
Int. J. Heat Mass Transfer
,
114
, pp.
105
114
.10.1016/j.ijheatmasstransfer.2017.06.025
13.
Bird
,
E.
,
Gutierrez Plascencia
,
J.
, and
Liang
,
Z.
,
2020
, “
Thermal Transport Across the Interface Between Liquid n -Dodecane and Its Own Vapor: A Molecular Dynamics Study
,”
J. Chem. Phys.
,
152
(
18
), p.
184701
.10.1063/1.5144279
14.
Chandra
,
A.
, and
Keblinski
,
P.
,
2020
, “
Investigating the Validity of Schrage Relationships for Water Using Molecular Dynamics Simulations
,”
J. Chem. Phys.
,
153
(
12
), p.
124505
.10.1063/5.0018726
15.
Sone
,
Y.
,
2007
,
Molecular Gas Dynamics Theory, Techniques, and Applications
,
Birkhauser
,
Boston, MA
.
16.
Meland
,
R.
, and
Ytrehus
,
T.
,
2003
, “
Evaporation and Condensation Knudsen Layers for Nonunity Condensation Coefficient
,”
Phys. Fluids
,
15
(
5
), pp.
1348
1350
.10.1063/1.1564097
17.
Meland
,
R.
,
2003
, “
Molecular Dynamics Simulation of the Inverted Temperature Gradient Phenomenon
,”
Phys. Fluids
,
15
(
10
), p.
3244
.10.1063/1.1604779
18.
Aoki
,
K.
, and
Cercignani
,
C.
,
1983
, “
Evaporation and Condensation on Two Parallel Plates at Finite Reynolds Numbers
,”
Phys. Fluids
,
26
(
5
), p.
1163
.10.1063/1.864277
19.
Plesset
,
M. S.
,
1952
, “
Note on the Flow of Vapor Between Liquid Surfaces
,”
J. Chem. Phys.
,
20
(
5
), pp.
790
793
.10.1063/1.1700568
20.
Shankar
,
P. N.
, and
Deshpande
,
M. D.
,
1991
, “
The Gas Dynamics of Massive Evaporation and Condensation
,”
Philos. Trans. R. Soc. London. Ser. A Phys. Eng. Sci.
,
335
(
1639
), pp.
487
511
.10.1098/rsta.1991.0058
21.
Ytrehus
,
T.
,
1997
, “
Molecular-Flow Effects in Evaporation and Condensation Phenomena
,”
Multiphase Sci. Technol.
,
9
(
3
), pp.
205
327
.10.1615/MultScienTechn.v9.i3.10
22.
Graur
,
I. A.
,
Gatapova
,
E. Y.
,
Wolf
,
M.
, and
Batueva
,
M. A.
,
2021
, “
Non-Equilibrium Evaporation: 1D Benchmark Problem for Single Gas
,”
Int. J. Heat Mass Transfer
,
181
, p.
121997
.10.1016/j.ijheatmasstransfer.2021.121997
23.
Nabavian
,
K.
, and
Bromley
,
L. A.
,
1963
, “
Condensation Coefficient of Water
,”
Chem. Eng. Sci.
,
18
(
10
), pp.
651
660
.10.1016/0009-2509(63)85035-6
24.
Carey
,
V. P.
,
1992
,
Liquid Vapor Phase Change Phenomena: An Introduction to the Thermophysics of Vaporization and Condensation Processes in Heat Transfer Equipment
,
Hemisphere Publishing
, Boca Raton, FL.
25.
Zhakhovskii
,
V. V.
, and
Anisimov
,
S. I.
,
1997
, “
Molecular-Dynamics Simulation of Evaporation of a Liquid
,”
J. Exp. Theor. Phys.
,
84
(
4
), pp.
734
745
.10.1134/1.558192
26.
Ishiyama
,
T.
,
Yano
,
T.
, and
Fujikawa
,
S.
,
2004
, “
Molecular Dynamics Study of Kinetic Boundary Condition at an Interface Between Argon Vapor and Its Condensed Phase
,”
Phys. Fluids
,
16
(
8
), pp.
2899
2906
.10.1063/1.1763936
27.
Ishiyama
,
T.
,
Yano
,
T.
, and
Fujikawa
,
S.
,
2004
, “
Molecular Dynamics Study of Kinetic Boundary Condition at an Interface Between a Polyatomic Vapor and Its Condensed Phase
,”
Phys. Fluids
,
16
(
12
), pp.
4713
4726
.10.1063/1.1811674
28.
Frezzotti
,
A.
, and
Barbante
,
P.
,
2017
, “
Kinetic Theory Aspects of Non-Equilibrium Liquid-Vapor Flows
,”
Mech. Eng. Rev.
,
4
(
2
), p.
16
.10.1299/mer.16-00540
29.
Tsuruta
,
T.
,
Tanaka
,
H.
, and
Masuoka
,
T.
,
1999
, “
Condensation/Evaporation Coefficient and Velocity Distributions at Liquid-Vapor Interface
,”
Int. J. Heat Mass Transfer
,
42
(
22
), pp.
4107
4116
.10.1016/S0017-9310(99)00081-2
30.
Kobayashi
,
K.
,
Hori
,
K.
,
Kon
,
M.
,
Sasaki
,
K.
, and
Watanabe
,
M.
,
2016
, “
Molecular Dynamics Study on Evaporation and Reflection of Monatomic Molecules to Construct Kinetic Boundary Condition in Vapor–Liquid Equilibria
,”
Heat Mass Transfer
,
52
(
9
), pp.
1851
1859
.10.1007/s00231-015-1700-6
31.
Skarbalius
,
G.
,
Džiugys
,
A.
,
Misiulis
,
E.
, and
Navakas
,
R.
,
2021
, “
Molecular Dynamics Study on Water Evaporation/Condensation Parameters
,”
Microfluid. Nanofluid.
,
25
(
10
), p.
81
.10.1007/s10404-021-02482-3
32.
Wilhelmsen
,
Ø.
,
Trinh
,
T. T.
,
Lervik
,
A.
,
Badam
,
V. K.
,
Kjelstrup
,
S.
, and
Bedeaux
,
D.
,
2016
, “
Coherent Description of Transport Across the Water Interface: From Nanodroplets to Climate Models
,”
Phys. Rev. E
,
93
(
3
), p.
032801
.10.1103/PhysRevE.93.032801
33.
Eames
,
I. W.
,
Marr
,
N. J.
, and
Sabir
,
H.
,
1997
, “
The Evaporation Coefficient of Water: A Review
,”
Int. J. Heat Mass Transfer
,
40
(
12
), pp.
2963
2973
.10.1016/S0017-9310(96)00339-0
34.
Marek
,
R.
, and
Straub
,
J.
,
2001
, “
Analysis of the Evaporation Coefficient and the Condensation Coefficient of Water
,”
Int. J. Heat Mass Transfer
,
44
(
1
), pp.
39
53
.10.1016/S0017-9310(00)00086-7
35.
PAUL
,
B.
,
1962
, “
Compilation of Evaporation Coefficients
,”
ARS J.
,
32
(
9
), pp.
1321
1328
.10.2514/8.6277
36.
Vaartstra
,
G.
,
Zhang
,
L.
,
Lu
,
Z.
,
Díaz-Marín
,
C. D.
,
Grossman
,
J. C.
, and
Wang
,
E. N.
,
2020
, “
Capillary-Fed, Thin Film Evaporation Devices
,”
J. Appl. Phys.
,
128
(
13
), p. 130901.10.1063/5.0021674
37.
Vincenti
,
W. G.
, and
Kruger
,
C. H.
,
1965
,
Introduction to Physical Gas Dynamics
,
Wiley
,
New York
.
38.
Cercignani
,
C.
,
2000
,
Rarefied Gas Dynamics: From Basic Concepts to Actual Calculations
,
Cambridge University Press
,
Cambridge, UK
.
39.
Umur
,
A.
, and
Griffith
,
P.
,
1965
, “
Mechanism of Dropwise Condensation
,”
ASME J. Heat Transfer-Trans. ASME
,
87
(
2
), pp.
275
282
.10.1115/1.3689090
40.
Faghri
,
A.
,
1995
,
Heat Pipe Science and Technology
,
Taylor & Francis
,
Washington, DC
.
41.
Wang
,
H.
,
Garimella
,
S. V.
, and
Murthy
,
J. Y.
,
2008
, “
An Analytical Solution for the Total Heat Transfer in the Thin-Film Region of an Evaporating Meniscus
,”
Int. J. Heat Mass Transfer
,
51
(
25–26
), pp.
6317
6322
.10.1016/j.ijheatmasstransfer.2008.06.011
42.
Sone
,
Y.
,
2000
, “
Kinetic Theoretical Studies of the Half-Space Problem of Evaporation and Condensation
,”
Transp. Theory Stat. Phys.
,
29
(
3–5
), pp.
227
260
.10.1080/00411450008205874
43.
Frezzotti
,
A.
,
2007
, “
A Numerical Investigation of the Steady Evaporation of a Polyatomic Gas
,”
Eur. J. Mech. B/Fluids
,
26
(
1
), pp.
93
104
.10.1016/j.euromechflu.2006.03.007
44.
Siewert
,
C. E.
, and
Thomas
,
J. R.
,
1973
, “
Half-Space Problems in the Kinetic Theory of Gases
,”
Phys. Fluids
,
16
(
9
), p.
1557
.10.1063/1.1694568
45.
Sone
,
Y.
,
Ohwada
,
T.
, and
Aoki
,
K.
,
1989
, “
Evaporation and Condensation on a Plane Condensed Phase: Numerical Analysis of the Linearized Boltzmann Equation for Hard‐Sphere Molecules
,”
Phys. Fluids A Fluid Dyn.
,
1
(
8
), pp.
1398
1405
.10.1063/1.857316
46.
Luikov
,
A. V.
,
Perelman
,
T. L.
, and
Anisimov
,
S. I.
,
1971
, “
Evaporation of a Solid Into Vacuum
,”
Int. J. Heat Mass Transfer
,
14
(
2
), pp.
177
184
.10.1016/0017-9310(71)90087-1
47.
Ytrehus
,
T.
, and
Alvestad
,
J. A.
,
1981
, “
A Mott-Smith Solution for Nonlinear Condensation
,”
Rarefied Gas Dynamics, Part I
, American Institute of Aeronautics and Astronautics,
New York
, pp.
330
345
.
48.
Shankar
,
P. N.
, and
Marble
,
F. E.
,
1971
, “
Kinetic Theory of Transient Condensation and Evaporation at a Plane Surface
,”
Phys. Fluids
,
14
(
3
), p.
510
.10.1063/1.1693464
49.
Bhatnagar
,
P. L.
,
Gross
,
E. P.
, and
Krook
,
M.
,
1954
, “
A Model for Collision Processes in Gases. I. Small Amplitude Processes in Charged and Neutral One-Component Systems
,”
Phys. Rev.
,
94
(
3
), pp.
511
525
.10.1103/PhysRev.94.511
50.
Welander
,
P.
,
1954
, “
On the Temperature Jump in a Rarefied Gas
,”
Ark. Fys.
,
7
, pp.
507
553
.https://www.osti.gov/biblio/4395580
51.
Aoki
,
K.
,
Sone
,
Y.
, and
Yamada
,
T.
,
1990
, “
Numerical Analysis of Gas Flows Condensing on Its Plane Condensed Phase on the Basis of Kinetic Theory
,”
Phys. Fluids A Fluid Dyn.
,
2
(
10
), pp.
1867
1878
.10.1063/1.857661
52.
Kogan
,
M. N.
, and
Makashev
,
N. K.
,
1974
, “
Role of the Knudsen Layer in the Theory of Heterogeneous Reactions and in Flows With Surface Reactions
,”
Fluid Dyn.
,
6
(
6
), pp.
913
920
.10.1007/BF01019794
53.
Bird
,
G. A.
,
1976
,
Molecular Gas Dynamics
,
Clarendon Press
,
Oxford, UK
.
54.
Bird
,
G. A.
,
1994
,
Molecular Gas Dynamics and the Direct Simulation of Gas Flows
,
Oxford University Press
,
New York
.
55.
Frezzotti
,
A.
, and
Ytrehus
,
T.
,
2006
, “
Kinetic Theory Study of Steady Condensation of a Polyatomic Gas
,”
Phys. Fluids
,
18
(
2
), p.
027101
.10.1063/1.2171231
56.
Borgnakke
,
C.
, and
Larsen
,
P. S.
,
1975
, “
Statistical Collision Model for Monte Carlo Simulation of Polyatomic Gas Mixture
,”
J. Comput. Phys.
,
18
(
4
), pp.
405
420
.10.1016/0021-9991(75)90094-7
57.
Holway
,
L. H.
,
1966
, “
New Statistical Models for Kinetic Theory: Methods of Construction
,”
Phys. Fluids
,
9
(
9
), p.
1658
.10.1063/1.1761920
58.
Lu
,
Z.
,
Kinefuchi
,
I.
,
Wilke
,
K. L.
,
Vaartstra
,
G.
, and
Wang
,
E. N.
,
2019
, “
A Unified Relationship for Evaporation Kinetics at Low Mach Numbers
,”
Nat. Commun.
,
10
(
1
), p.
2368
.10.1038/s41467-019-10209-w
59.
Ginoux
,
J. J.
,
1978
,
Two-Phase Flows and Heat Transfer With Application to Nuclear Reactor Design Problems
,
Hemisphere Publishing Corporation, Washington, DC
.
60.
Stultz
,
S. C.
, and
Kitto
,
J. B.
, eds.,
1992
,
Steam/Its Generation and Use
, The Babcock & Wilcox,
Barberton
,
OH
.
61.
Spiegler
,
K. S.
, and
El-Sayed
,
Y. M.
,
1994
,
A Desalination Primer
,
Balaban Desalination Publishers
, Santa Maria Imbaro, Italy.
62.
Davis
,
E. J.
,
2006
, “
A History and State-of-the-Art of Accommodation Coefficients
,”
Atmos. Res.
,
82
(
3–4
), pp.
561
578
.10.1016/j.atmosres.2006.02.013
63.
Tamir
,
A.
, and
Hasson
,
D.
,
1971
, “
Evaporation and Condensation Coefficient of Water
,”
Chem. Eng. J.
,
2
(
3
), pp.
200
211
.10.1016/0300-9467(71)80017-5
64.
Smith
,
J. D.
,
Cappa
,
C. D.
,
Drisdell
,
W. S.
,
Cohen
,
R. C.
, and
Saykally
,
R. J.
,
2006
, “
Raman Thermometry Measurements of Free Evaporation From Liquid Water Droplets
,”
J. Am. Chem. Soc.
,
128
(
39
), pp.
12892
12898
.10.1021/ja063579v
65.
Li
,
Y.
,
Chen
,
H.
,
Xiao
,
S.
,
Alibakhshi
,
M. A.
,
Lo
,
C.-W.
,
Lu
,
M.-C.
, and
Duan
,
C.
,
2019
, “
Ultrafast Diameter-Dependent Water Evaporation From Nanopores
,”
ACS Nano
,
13
(
3
), pp.
3363
3372
.10.1021/acsnano.8b09258
66.
Meland
,
R.
,
Frezzotti
,
A.
,
Ytrehus
,
T.
, and
Hafskjold
,
B.
,
2004
, “
Nonequilibrium Molecular-Dynamics Simulation of Net Evaporation and Net Condensation, and Evaluation of the Gas-Kinetic Boundary Condition at the Interphase
,”
Phys. Fluids
,
16
(
2
), pp.
223
243
.10.1063/1.1630797
67.
Mott-Smith
,
H. M.
,
1951
, “
The Solution of the Boltzmann Equation for a Shock Wave
,”
Phys. Rev.
,
82
(
6
), pp.
885
892
.10.1103/PhysRev.82.885
68.
Cercignani
,
C.
,
1980
, “
Strong Evaporation of a Polyatomic Gas
,”
Rarefied Gas Dynamics; International Symposium, 12th, Charlottesville, VA, July 7–11, Technical Papers. Part 1, American Institute of Aeronautics and Astronautics, New York, pp. 305–320.
69.
Kogan
,
M. N.
,
1992
, “
Kinetic Theory in Aerothermodynamics
,”
Prog. Aerosp. Sci.
,
29
(
4
), pp.
271
354
.10.1016/0376-0421(92)90007-5
70.
Abramov
,
A. A.
, and
Kogan
,
M. N.
,
1984
, “
Conditions for Supersonic Condensation of Gas
,”
Dokl. Akad. Nauk SSSR
,
278
(
5
), pp.
1078
1081
.http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=dan&paperid=46745&option_lang=eng
71.
Aoki
,
K.
,
Nishino
,
K.
,
Sone
,
Y.
, and
Sugimoto
,
H.
,
1991
, “
Numerical Analysis of Steady Flows of a Gas Condensing on or Evaporating From Its Plane Condensed Phase on the Basis of Kinetic Theory: Effect of Gas Motion Along the Condensed Phase
,”
Phys. Fluids A Fluid Dyn.
,
3
(
9
), pp.
2260
2275
.10.1063/1.857907
72.
Chu
,
C. K.
,
1965
, “
Kinetic-Theoretic Description of the Formation of a Shock Wave
,”
Phys. Fluids
,
8
(
1
), p.
12
.10.1063/1.1761077
73.
Ytrehus
,
T.
,
1983
, “
Asymmetries in Evaporation and Condensation Knudsen Layer Problems
,”
Phys. Fluids
,
26
(
4
), p.
939
.10.1063/1.864244
You do not currently have access to this content.